含复阻尼振动系统的逐步积分法稳定性研究
|
摘要
逐步积分法对于黏滞阻尼振动系统是条件稳定的,但用于求解复阻尼振动方程时,无一例外地出现不稳定现象。首先,针对复阻尼理论应遵循对偶复化的原则,给出Newmark-β法求解复阻尼振动方程的逐步列式。然后,根据理论推导,探讨Newmark-β法求解黏滞阻尼系统的稳定性条件。最后,通过理论分析和数值试验表明,对于复阻尼振动系统,求解不稳定的原因并非由于数值方法的不稳定,而是源于复阻尼本构方程本身的病态。稳定性条件与结构自振周期、荷载持续时间以及复阻尼常数的大小有关,当荷载持续时间与结构自振周期的比值小于某临界稳定值时,可以得到稳定的数值解。
Step-by-step method is conditionally stable for viscous damping systems.However,it is instable without exception when used in complex damping vibration equations.First,based on the dual principle of complex damping theory,the step-by-step equations of Newmark-β method were presented to solve the stability problems of vibration systems with complex damping.Then,the stability conditions of Newmark-β method in solving viscous damping systems were studied.Finally,by means of theoretical and numerical analysis,it was found that for complex damping vibration systems the instability of numerical solutions was not resulted from the algorithm itself but the divergent complex constitutive equation.The stability conditions of complex damping systems have relations with the period of structure vibration,the duration of exciting force and the value of the complex damping coefficient.A stable numerical solution can be obtained when the practical ratio of the duration of exciting force to the period of structure is smaller than the critical stable ratio.
Step-by-step method is conditionally stable for viscous damping systems.However,it is instable without exception when used in complex damping vibration equations.First,based on the dual principle of complex damping theory,the step-by-step equations of Newmark-β method were presented to solve the stability problems of vibration systems with complex damping.Then,the stability conditions of Newmark-β method in solving viscous damping systems were studied.Finally,by means of theoretical and numerical analysis,it was found that for complex damping vibration systems the instability of numerical solutions was not resulted from the algorithm itself but the divergent complex constitutive equation.The stability conditions of complex damping systems have relations with the period of structure vibration,the duration of exciting force and the value of the complex damping coefficient.A stable numerical solution can be obtained when the practical ratio of the duration of exciting force to the period of structure is smaller than the critical stable ratio.
引文
[1]克拉夫R.结构动力学[M].第二版.王光远,译.北京:高等教育出版社,2006
[2]王勋成.有限单元法[M].北京:清华大学出版社,2003
[3]朱敏,朱镜清.逐步积分法求解复阻尼结构运动方程的稳定性问题[J].地震工程与工程振动,2001,21(4):59-62(Zhu Min,Zhu Jingqing.Studies on stability ofstep-by-step methods under complex damping conditions[J].Earthquake Engineering and Engineering Vibration,2001,21(4):59-62(in Chinese))
[4]王建平,刘宏昭.含复阻尼振动系统的对偶原则及其数值方法研究[J].振动工程学报,2004,17(1):62-65(Wang Jianping,Liu Hongzhao.Dual principle of oscillationsystems with complex damping and its numerical method[J].Journal of Vibration Engineering,2004,17(1):62-65(in Chinese))
[5]何钟怡.复本构理论中的对偶原则[J].固体力学学报,1994,15(2):177-180(He Zhongyi.The dual principein theory of complex constitutive equations[J].ActaMechanica Solida Sinica,1994,15(2):177-180(in Chinese))
[6]朱镜清.关于复阻尼理论的两个基本问题[J].固体力学学报,1992,13(2):113-118(Zhu Jingqing.On the twobasicproblems of complex damping theory[J].Acta MechanicaSolida Sinica,1992,13(2):113-118(in Chinese))
[7]何钟怡.关于复阻尼理论的几点注记[J].地震工程与工程振动,2002,22(1):1-6(He Zhongyi.Some notes ontheory of complex damping[J].Earthquake Engineering andEngineering Vibration,2002,22(1):1-6(in Chinese))
[8]周正华,廖振鹏,丁海平.一种时域复阻尼本构方程[J].地震工程与工程振动,1999,19(2):37-44(ZhouZhenghua,Liao Zhenpeng,Ding Haiping.A time-domaincomplex-damping constitutive equation[J].EarthquakeEngineering and Engineering Vibration,1999,19(2):37-44(in Chinese))
[9]朱敏,朱镜清.复阻尼地震反应谱计算的再研究[J].地震工程与工程振动,2001,21(2):1-7,36-40(ZhuMin,Zhu Jingqing.Further studies on calculation ofcomplex damping seismic response spectra[J].EarthquakeEngineering and Engineering Vibration,2001,21(2):1-7,36-40(in Chinese))
[10]朱镜清,朱敏.复阻尼多自由度系统动力分析的模态叠加法[J].地震工程与工程振动,2004,24(1):55-58(Zhu Jingqing,Zhu Min.Two mode-superposition methodsfor dynamic analysis of MDOF systems with complexdamping characteristic[sJ].Earthquake Engineering andEngineering Vibration,2004,24(1):55-58(in Chinese))
[11]朱镜清.结构抗震分析原理[M].北京:地震出版社,2002
[12]李鹏,王元丰.黏性阻尼与复阻尼对钢筋混凝土框架结构地震响应影响的分析[J].地震工程与工程振动,2007,27(3):54-57(Li Peng,Wang Yuanfeng.Analysisof seismic responses of RC frames with viscous dampingor complex damping[J].Earthquake Engineering andEngineering Vibration,2007,27(3):54-57(in Chinese))
[2]王勋成.有限单元法[M].北京:清华大学出版社,2003
[3]朱敏,朱镜清.逐步积分法求解复阻尼结构运动方程的稳定性问题[J].地震工程与工程振动,2001,21(4):59-62(Zhu Min,Zhu Jingqing.Studies on stability ofstep-by-step methods under complex damping conditions[J].Earthquake Engineering and Engineering Vibration,2001,21(4):59-62(in Chinese))
[4]王建平,刘宏昭.含复阻尼振动系统的对偶原则及其数值方法研究[J].振动工程学报,2004,17(1):62-65(Wang Jianping,Liu Hongzhao.Dual principle of oscillationsystems with complex damping and its numerical method[J].Journal of Vibration Engineering,2004,17(1):62-65(in Chinese))
[5]何钟怡.复本构理论中的对偶原则[J].固体力学学报,1994,15(2):177-180(He Zhongyi.The dual principein theory of complex constitutive equations[J].ActaMechanica Solida Sinica,1994,15(2):177-180(in Chinese))
[6]朱镜清.关于复阻尼理论的两个基本问题[J].固体力学学报,1992,13(2):113-118(Zhu Jingqing.On the twobasicproblems of complex damping theory[J].Acta MechanicaSolida Sinica,1992,13(2):113-118(in Chinese))
[7]何钟怡.关于复阻尼理论的几点注记[J].地震工程与工程振动,2002,22(1):1-6(He Zhongyi.Some notes ontheory of complex damping[J].Earthquake Engineering andEngineering Vibration,2002,22(1):1-6(in Chinese))
[8]周正华,廖振鹏,丁海平.一种时域复阻尼本构方程[J].地震工程与工程振动,1999,19(2):37-44(ZhouZhenghua,Liao Zhenpeng,Ding Haiping.A time-domaincomplex-damping constitutive equation[J].EarthquakeEngineering and Engineering Vibration,1999,19(2):37-44(in Chinese))
[9]朱敏,朱镜清.复阻尼地震反应谱计算的再研究[J].地震工程与工程振动,2001,21(2):1-7,36-40(ZhuMin,Zhu Jingqing.Further studies on calculation ofcomplex damping seismic response spectra[J].EarthquakeEngineering and Engineering Vibration,2001,21(2):1-7,36-40(in Chinese))
[10]朱镜清,朱敏.复阻尼多自由度系统动力分析的模态叠加法[J].地震工程与工程振动,2004,24(1):55-58(Zhu Jingqing,Zhu Min.Two mode-superposition methodsfor dynamic analysis of MDOF systems with complexdamping characteristic[sJ].Earthquake Engineering andEngineering Vibration,2004,24(1):55-58(in Chinese))
[11]朱镜清.结构抗震分析原理[M].北京:地震出版社,2002
[12]李鹏,王元丰.黏性阻尼与复阻尼对钢筋混凝土框架结构地震响应影响的分析[J].地震工程与工程振动,2007,27(3):54-57(Li Peng,Wang Yuanfeng.Analysisof seismic responses of RC frames with viscous dampingor complex damping[J].Earthquake Engineering andEngineering Vibration,2007,27(3):54-57(in Chinese))
版权所有：© 2023 中国地质图书馆 中国地质调查局地学文献中心